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On fine differentiability properties of horizons and applications to Riemannian geometry

✍ Scribed by Piotr T. Chruściel; Joseph H.G. Fu; Gregory J. Galloway; Ralph Howard

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
108 KB
Volume
41
Category
Article
ISSN
0393-0440

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✦ Synopsis

We study fine differentiability properties of horizons. We show that the set of end points of generators of an n-dimensional horizon H (which is included in an (n + 1)-dimensional space-time M) has vanishing n-dimensional Hausdorff measure. This is proved by showing that the set of end points of generators at which the horizon is differentiable has the same property. For 1 ≤ k ≤ n + 1, we show (using deep results of Alberti) that the set of points where the convex hull of the set of generators leaving the horizon has dimension k is "almost a C 2 manifold of dimension n + 1 -k": it can be covered, up to a set of vanishing (n + 1 -k)-dimensional Hausdorff measure, by a countable number of C 2 manifolds. We use our Lorentzian geometry results to derive information about the fine differentiability properties of the distance function and the structure of cut loci in Riemannian geometry.

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